Algebraic & Geometric Topology

Conjugacy of $2$–spherical subgroups of Coxeter groups and parallel walls

Pierre-Emmanuel Caprace

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Let (W,S) be a Coxeter system of finite rank (ie |S| is finite) and let A be the associated Coxeter (or Davis) complex. We study chains of pairwise parallel walls in A using Tits’ bilinear form associated to the standard root system of (W,S). As an application, we prove the strong parallel wall conjecture of G Niblo and L Reeves [J Group Theory 6 (2003) 399–413]. This allows to prove finiteness of the number of conjugacy classes of certain one-ended subgroups of W, which yields in turn the determination of all co-Hopfian Coxeter groups of 2–spherical type.

Article information

Algebr. Geom. Topol., Volume 6, Number 4 (2006), 1987-2029.

Received: 2 August 2005
Revised: 31 August 2006
Accepted: 4 October 2006
First available in Project Euclid: 20 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 20F5
Secondary: 20F65: Geometric group theory [See also 05C25, 20E08, 57Mxx] 20F67: Hyperbolic groups and nonpositively curved groups 51F15: Reflection groups, reflection geometries [See also 20H10, 20H15; for Coxeter groups, see 20F55]

Coxeter group conjugacy class Hopfian group hyperbolic triangle parallel walls


Caprace, Pierre-Emmanuel. Conjugacy of $2$–spherical subgroups of Coxeter groups and parallel walls. Algebr. Geom. Topol. 6 (2006), no. 4, 1987--2029. doi:10.2140/agt.2006.6.1987.

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